Question

Assertion (A): If the volumes of two spheres are in the ratio 125 : 64 then the ratio of their surface areas is 25 : 16.

Reason (R): If the volume and surface area of a sphere are numerically equal then the radius of the sphere is 3 units.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. Since sphere volume is proportional to the cube of the radius, the volume ratio $$125 : 64$$ gives the radii ratio $$5 : 4$$. The surface-area ratio is therefore $$5^{2} : 4^{2} = 25 : 16$$.

Step 2 – Reason: R is true. Equating the numerical values $$\frac{4}{3}\pi r^{3} = 4\pi r^{2}$$ gives $$r = 3$$ units.

Step 3 – Link: The radius obtained when a sphere’s volume and surface area are numerically equal is unrelated to the ratio of the surface areas of two spheres.